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|
; Copyright (C) 2020, Regents of the University of Texas
; Written by Matt Kaufmann and J Moore
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Many thanks to ForrestHunt, Inc. for supporting the preponderance of this
; work, and for permission to include it here.
(in-package "ACL2")
(include-book "projects/apply/loop" :dir :system)
; -----------------------------------------------------------------
; Most of our test functions below return 0 and use the loop$ accumulator SUM.
; They were written to develop the nugget idea, which initially only handled
; SUM loop$s. But z0 is a simple loop$ recursive that uses COLLECT, just to
; test the generalization of the nugget idea.
(defun$ z0 (x)
(declare (xargs :loop$-recursion t :measure (acl2-count x)))
(cond ((atom x)
(if (natp x)
(if (zp x)
0
(+ 1 (z0 (- x 1))))
x))
((true-listp x)
(loop$ for e in x collect (z0 e)))
(t x)))
(definductor z0)
(defthm z0-thm
(implies (warrant z0)
(and (equal (z0 x) x)
(implies (true-listp x)
(equal (loop$ for e in x collect (z0 e)) x))))
:rule-classes nil)
; -----------------------------------------------------------------
; Fn0
; Features
; (1) recursion both outside and inside a loop$
; (2) case analysis inside the loop$ leading to two kinds of recursions
(defthm acl2-count-unary--
(implies (integerp x)
(equal (acl2-count (- x))
(acl2-count x))))
(defun$ fn0 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
0
(fn0 (- x 1))))
((true-listp x)
(loop$ for e in x sum
(if (and (integerp e) (< e 0))
(fn0 (- e))
(fn0 e))))
(t 0)))
(definductor fn0)
(defthm fn0-thm
(implies (warrant fn0)
(and (equal (fn0 x) 0)
(implies (true-listp x) ; <--- this hyp is optional
(equal (loop$ for e in x sum
(if (and (integerp e) (< e 0))
(fn0 (- e))
(fn0 e)))
0))))
:rule-classes nil)
; -----------------------------------------------------------------
; Fn1
; Features: Like fn0 but the target is cddr'd.
(defun$ fn1 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
0
(fn1 (- x 1))))
((true-listp x)
(loop$ for e in (cddr x) sum
(if (and (integerp e) (< e 0))
(fn1 (- e))
(fn1 e))))
(t 0)))
(definductor fn1)
(defthm fn1-thm
(implies (warrant fn1)
(and (equal (fn1 x) 0)
(implies (true-listp x) ; <--- this hyp is optional
(equal (loop$ for e in x sum
(if (and (integerp e) (< e 0))
(fn1 (- e))
(fn1 e)))
0))))
:rule-classes nil)
; -----------------------------------------------------------------
; Fn2
; Features: The simplest nested loop$.
(defun$ fn2 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
0
(fn2 (- x 1))))
((true-listp x)
(loop$ for e in x sum
(loop$ for d in e sum (fn2 d))))
(t 0)))
(definductor fn2)
(defthm fn2-thm
(implies (warrant fn2 sum$)
(and (equal (fn2 x) 0)
(implies (true-listp x) ; <--- this hyp is optional
(equal (loop$ for e in x sum
(loop$ for d in e sum
(fn2 d)))
0))
(equal (loop$ for d in x sum
(fn2 d))
0)))
:rule-classes nil)
; -----------------------------------------------------------------
; Fn3
; Features: a nested loop where the inner target is cddr'd
(defun$ fn3 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
0
(fn3 (- x 1))))
((true-listp x)
(loop$ for e in x sum
(loop$ for d in (cddr e) sum (fn3 d))))
(t 0)))
(definductor fn3)
(defthm fn3-thm
(implies (warrant fn3 sum$)
(and (equal (fn3 x) 0)
(equal (loop$ for e in x sum
(loop$ for d in (cddr e) sum
(fn3 d)))
0)
(equal (loop$ for d in x sum
(fn3 d))
0)
))
:rule-classes nil)
; -----------------------------------------------------------------
; Fn4
; Features: A nested loop with cddr, interior cases, and more recursions.
(defun$ fn4 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
0
(fn4 (- x 1))))
((true-listp x)
(loop$ for e in x sum
(+ (fn4 e)
(loop$ for d in (cddr e) sum
(if (and (integerp d) (< d 0))
(fn4 (- d))
(fn4 d))))))
(t 0)))
(definductor fn4)
(defthm fn4-thm
(implies (warrant fn4 sum$)
(and (equal (fn4 x) 0)
(equal (loop$ for e in x sum
(+ (fn4 e)
(loop$ for d in (cddr e) sum
(if (and (integerp d) (< d 0))
(fn4 (- d))
(fn4 d)))))
0)
(equal (loop$ for d in x sum
(if (and (integerp d) (< d 0))
(fn4 (- d))
(fn4 d)))
0)))
:rule-classes nil)
; Time: 4.21 seconds (prove: 4.19, print: 0.02, other: 0.00)
; -----------------------------------------------------------------
; Fn5
; Features: Tripply nested loops with cddr and caar
(defun$ fn5 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
0
(fn5 (- x 1))))
((true-listp x)
(loop$ for e in x sum
(+ (fn5 e)
(loop$ for c in (cddr e) sum
(loop$ for d in (cddr c) sum
(if (and (integerp d) (< d 0))
(fn5 (- d))
(fn5 d)))))))
(t 0)))
(definductor fn5)
(defthm fn5-thm
(implies (warrant fn5 sum$)
(and (equal (fn5 x) 0)
(equal (loop$ for e in x sum
(+ (fn5 e)
(loop$ for c in (cddr e) sum
(loop$ for d in (cddr c) sum
(if (and (integerp d) (< d 0))
(fn5 (- d))
(fn5 d))))))
0)
(equal (loop$ for c in x sum
(loop$ for d in (cddr c) sum
(if (and (integerp d) (< d 0))
(fn5 (- d))
(fn5 d))))
0)
(equal (loop$ for d in x sum
(if (and (integerp d) (< d 0))
(fn5 (- d))
(fn5 d)))
0)))
:rule-classes nil)
;-----------------------------------------------------------------
; z1
; Features: Extra formals. However, to prevent fn6 from using fancy scions the
; extra actuals in loop$ recursive calls are constants!
(defun$ z1 (a x b)
(declare (xargs :loop$-recursion t :measure (acl2-count x)))
(cond ((atom x)
(if (natp x)
(if (zp x)
0
(+ 1 (z1 (+ 1 a) (- x 1) (* 2 b))))
x))
((true-listp x)
(loop$ for e in x collect (z1 0 e 1)))
(t (* (+ a b) x))))
(definductor z1)
(defthm z1-thm
(implies (warrant z1)
(and (implies (natp x) (equal (z1 a x c) x))
(implies (nat-listp x)
(equal (loop$ for e in x collect (z1 0 e 1)) x))))
:rule-classes nil)
;-----------------------------------------------------------------
; Fn6
; Features: Our first fancy loop$.
(defun$ fn6 (x c)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((natp x)
(if (equal x 0)
c
(fn6 (- x 1) c)))
((true-listp x)
(loop$ for e in x sum
(if (and (integerp e) (< e 0))
(fn6 (- e) c)
(fn6 e c))))
(t c)))
(definductor fn6)
(defthm fn6-thm
(implies (warrant fn6)
(and (implies (natp x) (equal (fn6 x 0) 0))
(implies (nat-listp x)
(equal (loop$ for e in x sum (fn6 e 0)) 0))))
:rule-classes nil)
;-----------------------------------------------------------------
; Fn7
; Features: Simultaneous sweep over two vars.
(defun$ fn7 (x c y)
(declare (xargs :loop$-recursion t
:measure (+ (acl2-count x)
(acl2-count y))))
(cond
((natp x)
(if (equal x 0)
c
(fn7 (- x 1) c y)))
((natp y)
(if (equal y 0)
c
(fn7 x c (- y 1))))
(t (loop$ for e in x as d in y
sum
(cond ((and (consp e)
(consp d))
(fn7 e c d))
(t c))))))
(definductor fn7)
(defthm fn7-thm
(implies (warrant fn7)
(and (implies (natp x) (equal (fn7 x 0 y) 0))
(implies (natp y) (equal (fn7 x 0 y) 0))
(implies (equal c 0)
(equal (loop$ for e in x as d in y
sum
(cond ((and (consp e)
(consp d))
(fn7 e c d))
(t c)))
0))))
:rule-classes nil)
; -----------------------------------------------------------------
; Z2
; Feature: Relating a loop$ recursive function to a mutually recursive one.
; This function returns 0 but is sort of an extreme simplification of
; pseudo-termp-type recursion.
(defun$ z2 (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond ((atom x)
0)
(t (loop$ for e in (cdr x) sum (z2 e)))))
(definductor z2)
(defthm z2-thm
(implies (warrant z2)
(and (equal (z2 x) 0)
(equal (loop$ for e in x sum (z2 e)) 0)))
:rule-classes nil)
(mutual-recursion
(defun mr-z2 (x)
(declare (xargs :measure (acl2-count x)))
(if (atom x)
0
(mr-z2-list (cdr x))))
(defun mr-z2-list (x)
(declare (xargs :measure (acl2-count x)))
(if (endp x)
0
(+ (fix (mr-z2 (car x)))
(mr-z2-list (cdr x)))))
)
(defthm mr-z2-is-z2
(implies (warrant z2)
(and (equal (z2 x) (mr-z2 x))
(equal (loop$ for e in x sum (z2 e))
(mr-z2-list x))))
:rule-classes nil)
; -----------------------------------------------------------------
; Copy-nat-tree
; Features: a little closer to pseudo-termp and exploring mutual recursion
(defun$ nat-treep (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((atom x) (natp x))
(t (and (true-listp x)
(eq (car x) 'NATS)
(loop$ for e in (cdr x) always (nat-treep e))))))
(defthm examples-of-nat-treep
(implies
(warrant nat-treep)
(and (equal (nat-treep '(nats
(nats 1 2 3)
4
(nats 5 (nats 6 7 8) 9))) t)
(equal (nat-treep '(nats (nats 1 2 3) bad)) nil)))
:rule-classes nil)
(definductor nat-treep)
(defun$ copy-nat-tree (x)
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond
((atom x)
(if (natp x)
(if (equal x 0)
0
(+ 1 (copy-nat-tree (- x 1))))
x))
(t (cons 'nats
(loop$ for e in (cdr x) collect (copy-nat-tree e))))))
(definductor copy-nat-tree)
(defthm copy-nat-tree-copies
(implies (warrant nat-treep copy-nat-tree)
(and (implies (nat-treep x) (equal (copy-nat-tree x) x))
(implies (and (true-listp x)
(loop$ for e in x always (nat-treep e)))
(equal (loop$ for e in x collect (copy-nat-tree e))
x))))
:rule-classes nil)
(mutual-recursion
(defun mr-copy-nat-tree (x)
(cond
((atom x)
(if (natp x)
(if (equal x 0)
0
(+ 1 (mr-copy-nat-tree (- x 1))))
x))
(t (cons 'nats
(mr-copy-nat-tree-list (cdr x))))))
(defun mr-copy-nat-tree-list (x)
(cond
((endp x) nil)
(t (cons (mr-copy-nat-tree (car x))
(mr-copy-nat-tree-list (cdr x)))))))
(defthm mr-copy-nat-tree-copies
(implies (warrant nat-treep)
(and (implies (nat-treep x)
(equal (mr-copy-nat-tree x) x))
(implies (and (true-listp x)
(loop$ for e in x always (nat-treep e)))
(equal (mr-copy-nat-tree-list x) x))))
:hints (("Goal" :induct (copy-nat-tree x)))
:rule-classes nil)
(defthm copy-nat-tree-is-mr-copy-nat-tree
(implies (warrant nat-treep copy-nat-tree)
(and (implies (nat-treep x)
(equal (copy-nat-tree x)
(mr-copy-nat-tree x)))
(implies (and (true-listp x)
(loop$ for e in x always (nat-treep e)))
(equal (loop$ for e in x collect (copy-nat-tree e))
(mr-copy-nat-tree-list x)))))
:rule-classes nil)
; -----------------------------------------------------------------
; Pstermp
; Features: trying the methodology above.
(defun$ pstermp (x)
; This is the ACL2 built-in pseudo-termp, expressed with loop$
(declare (xargs :loop$-recursion t
:measure (acl2-count x)))
(cond ((atom x) (symbolp x))
((eq (car x) 'quote)
(and (consp (cdr x))
(null (cdr (cdr x)))))
((not (true-listp x)) nil)
((loop$ for e in (cdr x) always (pstermp e))
(or (symbolp (car x))
(and (true-listp (car x))
(equal (length (car x)) 3)
(eq (car (car x)) 'lambda)
(symbol-listp (cadr (car x)))
(pstermp (caddr (car x)))
(equal (length (cadr (car x)))
(length (cdr x))))))
(t nil)))
(definductor pstermp)
(defthm pstermp-is-pseudo-termp
(implies (warrant pstermp)
(and (equal (pstermp x) (pseudo-termp x))
(equal (and (true-listp x)
(loop$ for e in x always (pstermp e)))
(pseudo-term-listp x))))
:rule-classes nil)
|
